Edge crossings in random linear arrangements
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In spatial networks, vertices are arranged in some space and edges may cross. Here we consider the particular case of arranging vertices in a 1-dimensional lattice, where edges may cross when drawn above the vertex sequence as it happens in linguistic and biological networks. Here we investigate the distribution of edge crossings under the null hypothesis of a uniformly random arrangement of the vertices. We generalize the existing formula for the expectation of this number in trees to any network and derive a general expression for the variance of the number of crossings relying on a novel characterization of the algebraic structure of that variance in an arbitrary space. We provide compact formulae for the expectation and the variance in complete graphs, cycle graphs, one-regular graphs and various kinds of trees (star trees, quasi-star trees and linear trees). In these networks, the scaling of expectation and variance as a function of network size is asymptotically power-law-like. Our work paves the way for further research and applications in 1-dimension or investigating the distribution of the number of crossings in lattices of higher dimension or other embeddings.
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